# Sunday Afternoon Maths III

## Twenty

How many three digit integers are there for which the product of the digits is 20?

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There are two ways to make 20 by multiplying three digits: \(2\times 2\times 5\) and \(1\times 4\times 5\). Listing all the possible orderings of these, we have:

145

154

415

451

514

541

225

252

522

Therefore, there are 9 different three digit numbers where the product of the digits is 20.

#### Extension

How many 4 digit numbers are there where the product of the digits is 20?

5 digit?

\(n\) digit?

## Colliding parallel people

If two people stand 1km apart and walk in the same direction, how far will the have to walk until they collide due to the curvature of the Earth? (diameter of Earth = 12,742km)

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By walking in a straight line, the people will follow a

great circle. They begin by travelling parallel to each other, so the 1km they are apart at the start is the furthest they will be apart. They will be 1km apart again after walking half way around the world and will collide at one quarter and three quarters of the way around.

Hence, each person will walk one quarter of the circumference of the Earth before colliding, which will be 10,018km.

Another way to see that this is the answer is to assume that the people stand on the equator and walk North. They will meet at the North Pole, one quarter of the way round.

#### Extension

If two people stand 10km apart and walk in the same direction, how far will the have to walk until they collide due to the curvature of the Earth? (diameter of Earth = 12,742km)